The project explored the use of geometric algebraic techniques in quantum computing and, in particular, NMR ensemble quantum computing. Novel mathematical techniques were developed to analyze and manipulate the basic quantum computing gates. These methods lend themselves to an algebraic scheme for developing NMR pulse sequences to implement the various quantum computing logic operations. The analysis allows for great flexibility in designing sequences. We can easily tailor the sequences for the specifics of the molecules (coupling, resonance frequencies etc) used to perform the logic operations. Further algebraic methods were developed that helped in realizing the first explicit simulation of one quantum system by another, namely, the simulation of harmonic and anharmonic oscillators on a two-spin 1/2 system. For any desired hamiltonian that we wish to simulate, the scheme allows one to easily derive the effective hamiltonian in the simulating system. This is complemented by the average hamiltonian theory of Waugh to obtain the desired pulse sequences necessary in an NMR implementation.